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A260550 a(n) is the number of 2 X 2 matrices with entries in {1, ..., n} that are not the product of two 2 X 2 positive integer matrices.

%I A260550 #32 Jan 01 2025 22:20:50
%S A260550 1,15,75,237,559,1157,2055,3471,5449,8131,11633,16361,22041,29349,
%T A260550 38329,48839,61325,76479,93957,114717,138041,164153,194505,229625,
%U A260550 268259,311031,359719,413245,472145,537835,608837,688121,774877,867549,971403,1080637,1198233,1326059,1467029,1617451,1777881,1948219,2132381,2329081,2539351
%N A260550 a(n) is the number of 2 X 2 matrices with entries in {1, ..., n} that are not the product of two 2 X 2 positive integer matrices.
%C A260550 a(n) <= A000583(n), which is the number of 2 X 2 matrices with entries in {1, ..., n}.
%C A260550 a(n) >= A005917(n), which is the number of 2 X 2 matrices with entries in {1, ..., n} that contain the element 1. All such matrices are not decomposable as a product of 2 X 2 positive integer matrices.
%C A260550 This definition is a generalization of the notion of prime numbers to the family of 2 X 2 positive integer matrices. Since the matrices do not contain 0, max(A*B) > max(A) and max(A*B) > max(B). Thus, for every matrix there is a finite number of possible decompositions to check.
%H A260550 Michael S. Branicky, <a href="/A260550/b260550.txt">Table of n, a(n) for n = 1..60</a>
%H A260550 Michael S. Branicky, <a href="/A260550/a260550.py.txt">Python program</a>
%H A260550 Aldo González Lorenzo, <a href="/A260550/a260550.txt">Scilab function for computing this sequence</a>
%H A260550 P. F. Rivett and N. I. P. Mackinnon, <a href="http://www.jstor.org/stable/3616179">Prime Matrices</a>, The Mathematical Gazette, Vol. 70, No. 454 (Dec., 1986), pp. 257-259.
%e A260550 The matrix [2,2;3,3] is decomposable: [2,2;3,3] = [1,1;1,2] * [1,1;1,1]. However, the matrix [2,3;3;2] is not decomposable.
%o A260550 (Python) # See Branicky link.
%Y A260550 Cf. A000583, A005917.
%K A260550 nonn,hard
%O A260550 1,2
%A A260550 _Aldo González Lorenzo_, Jul 29 2015